Usually it's just a difference in preferred terminology. While there are a few schools out there that do award a degree known as the Associate Degree in Nursing, usually ADN is a general term for a nursing degree that may actually be an Associate of Science, Associate of Applied Science, or even an Associate of Arts.
Some schools offer multiple associate degrees in nursing. Mine offers an Associate of Applied Science and an Associate of Science. Students with more elective credits in the arts and sciences go with the AS, while students who have credits in allied health or other applied professional fields may fit better into the AAS plan. In the end, the nursing courses are the same for us and we frequently refer to each other as all being "in the ADN program."
Does that make sense?